Less Men, means more Days {Indirect Proportion}

Let the number of days be x then,

27 : 36 :: 18 : x

[Please pay attention, we have written 27 : 36 rather than 36 : 27, in indirect proportion, if you get it then chain rule is clear to you :)]                                

=>27x = 36 * 18

=> x = 24

 So 24 days will be required to get work done by 27 men.

Quarter of Kg means 250 gm

Less weight, less price (Direct Proportion)

So,

250 : 200 :: 60 : x => x = 48

So 200 gm will cost 48 paise.

If we consider the third term to be ?x?
The 15th term will be (x + 12d)
6th term will be (x + 3d)
11th term will be (x + 8d) and
13th term will be (x + 10d).
Thus, as per the given condition, 2x + 12d = 3x + 21d.Or x + 9d = 0.
x + 9d will be the 12th term.

Thus, 12th term of the A.P will be zero.

Let x men can do the in 12 days and the required number of days be z

More men, Less days     [Indirect Proportion]

Less work, Less days     [Direct Proportion  ]

  m e n 2 x : x w o r k 1 : 1 2 : : 12 : z

? 2 x × 1 × z = x × 1 2 × 12

  ? z = 3

1 2 3 4 5 6(Shankar) 7(Nitu) 8(Althaf) 9 10 11 12 13 14
Here, Althaf is 8th from front, Shankar is 9th from rear end and Nitu is between them
So minimum no. of boys standing in the queue = 14

Assume first child (the youngest) get = Rs. x

According to the question ;
each son having Rs. 30 more than the younger one

Second child will get = Rs. x + 30
Third child will get = Rs. x + 30 + 30 = x + 60
Forth child will get = Rs. x + 30 + 30 + 30 = x + 90
Fifth child will get = Rs. x + 30 + 30 + 30 + 30 = x + 120

Total amount they got = Rs. 2000

x + (x+30) + (x+60) + (x+90) + (x+120) = 2000
5x + 300 = 2000
5x = 1700
x = Rs. 340
So the youngest child will get Rs. 340.

Let the required no of hectares be x. Then 

More men, More hectares      (Direct proportion)

More days, More hectares      (Direct proportion)

  M e n 8 : 36 D a y s 24 : 30 ? ? 80 : x

=> ( 8 * 24 * x) = (36 * 30 * 80)

=>  x= 450

Let the required number of revolutions made by larger wheel be x.

Then, More cogs, Less revolutions (Indirect Proportion)

  14 : 6 :: 21 : x => 14 x x = 6 x 21

x=6*21/14  =>  x=9

After 10 days : 150 men had food for 35 days.

Suppose 125 men had food for x days.

Now, Less men, More days (Indirect Proportion)

? 125 : 150 :: 35 : x    ?   125 x x = 150 x 35

  ? x = 150 × 35 125 = 42

Cost of x metres = Rs. d.

Cost of 1 metre = Rs.(d/x)

Cost of y metres = Rs.[(d/x).y] = Rs.(yd/x)